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Optimality of the one step look-ahead stopping times

Published online by Cambridge University Press:  14 July 2016

M. Abdel-Hameed
M. Abdel-Hameed*
Affiliation:
University of North Carolina at Charlotte
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Abstract

The optimality of the one step look-ahead stopping rule is shown to hold under conditions different from those discussed by Chow, Robbins and Seigmund [5]. These results are corollaries of the following theorem: Let {Xn, n = 0, 1, …}; X0 = x be a discrete-time homogeneous Markov process with state space (E, ℬ). For any ℬ-measurable function g and α in (0, 1], define Aαg(x) = αExg(X1) – g(x) to be the infinitesimal generator of g. If τ is any stopping time satisfying the conditions: Ex[αNg(XN)I(τ > N)]→0 as as N → ∞, then Applications of the results are considered.

Keywords

OPTIMAL STOPPING TIMESMARKOV TIMESDISCRETE-TIME HOMOGENEOUS MARKOV PROCESSESNON-HOMOGENEOUS MARKOV PROCESSESINCREASING FUNCTIONSROOT-INCREASING FUNCTIONSINFINITESIMAL GENERATORS
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 162 - 169
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Abdel-Hameed, M. S. and Proschan, F. (1973) Non-stationary shock models. Stoch. Proc. Appl. 1, 383–404.Google Scholar
[2] Abdel-Hameed, M. S. and Proschan, F. (1975) Shock models with underlying birth process. J. Appl. Prob. 12, 18–28.Google Scholar
[3] Abdel-Hameed, M. S. and Proschan, F. (1975) Total positivity properties of generating functions. SIAM J. Math. Anal. 6, 81–84.Google Scholar
[4] Abdel-Hameed, M. S. and Proschan, F. (1976) Preservation of geometric properties under an integral transformation SIAM J. Math. Anal. 7, 359–365.Google Scholar
[5] Chow, Y. S., Robbins, H. and Seigmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin Company, Boston.Google Scholar
[6] Chung, K. L. (1974) A Course in Probability Theory. Academic Press, New York.Google Scholar
[7] Dynkin, E. B. (1965) Markov Processes. (English translation) Springer-Verlag, Berlin.Google Scholar
[8] Sirjaev, A. N. (1973) Statistical Sequential Analysis. (English translation) American Mathematical Society, Providence, R.I. Google Scholar